Existence of counterexamples where the PRAX algorithm fails

Determine whether there exist instances of the PRAX framework’s universality or emptiness problems for which the algorithm A(σ, ε) fails; specifically, identify a subset description σ and tolerance ε such that σ is not ε-close to being universal (or not ε-close to being empty) relative to the associated distribution, yet A(σ, ε) returns true.

Background

The authors define failure for a PRAX algorithm A(σ, ε) as returning true even though the instance σ is not ε-close to universal or empty with respect to the chosen distribution.

They report that prior work on NFA universality found no failing examples and that, despite their efforts, they have not found any instances where the algorithm fails, which motivates the explicit question of whether such counterexamples exist.